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Singquandle shadows and singular knot invariants

Published online by Cambridge University Press:  24 September 2021

Jose Ceniceros ,
Indu R. Churchill  and
Mohamed Elhamdadi
Jose Ceniceros
Affiliation:
Department of Mathematics and Statistics, Hamilton College, College Hill Rd., NY13323, USA e-mail: jcenicer@hamilton.edu
Indu R. Churchill
Affiliation:
Mathematics Department, State University of New York at Oswego, 7060 NY-104, NY13126, USA e-mail: indurasika.churchill@oswego.edu
Mohamed Elhamdadi*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., FL33620, USA
*
e-mail: emohamed@usf.edu
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Abstract

We introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.

Keywords

Quandle polynomialsingular knots and linkssingquandle polynomial
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 770 - 787
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Copyright
© Canadian Mathematical Society 2021

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References

Bataineh, K., Elhamdadi, M., Hajij, M., and Youmans, W., Generating sets of Reidemeister moves of oriented singular links and quandles. J. Knot Theory Ramificat. 27(2018), no. 14, 1850064, 15.10.1142/S0218216518500645CrossRefGoogle Scholar
Birman, J. S. and Lin, X.-S., Knot polynomials and Vassiliev’s invariants. Invent. Math. 111(1993), no. 2, 225–270.CrossRefGoogle Scholar
Carter, J. S., Elhamdadi, M., and Saito, M., Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles. Fund. Math. 184(2004), 31–54.CrossRefGoogle Scholar
Carter, J. S., Jelsovsky, D., Kamada, S., Langford, L., and Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355(2003), no. 10, 3947–3989. https://doi.org/10.1090/S0002-9947-03-03046-0 CrossRefGoogle Scholar
Ceniceros, J., Churchill, I. R., and Elhamdadi, M., Polynomial invariants of singular knots and links. J. Knot Theory Ramificat. 30(2021), no. 1, Paper No. 2150003, 17. https://doi.org/10.1142/S0218216521500036 CrossRefGoogle Scholar
Ceniceros, J., Churchill, I. R., Elhamdadi, M., and Hajij, M., Cocycle invariants and oriented singular knots. Mediterr. J. Math. 18(2021), no. 5, 217. https://doi.org/10.1007/s00009-021-01867-6 CrossRefGoogle Scholar
Ceniceros, J. and Nelson, S., Virtual Yang-Baxter cocycle invariants. Trans. Amer. Math. Soc. 361(2009), no. 10, 5263–5283.CrossRefGoogle Scholar
Chang, W. and Nelson, S., Rack shadows and their invariants. J. Knot Theory Ramificat. 20(2011), no. 9, 1259–1269. https://doi.og/10.1142/S0218216511009315 CrossRefGoogle Scholar
Churchill, I. R. U., Elhamdadi, M., Hajij, M., and Nelson, S., Singular knots and involutive quandles. J. Knot Theory Ramificat. 26(2017), no. 14, 1750099, 14 pp.CrossRefGoogle Scholar
Elhamdadi, M., Distributivity in quandles and quasigroups. In: Algebra, geometry and mathematical physics, Springer Proc. Math. Stat., 85, Springer, Heidelberg, 2014, pp. 325–340.CrossRefGoogle Scholar
Elhamdadi, M., Fernando, N., and Tsvelikhovskiy, B., Ring theoretic aspects of quandles. J. Algebra 526(2019), 166–187.CrossRefGoogle Scholar
Elhamdadi, M. and Moutuou, E.-k. M., Finitely stable racks and rack representations. Comm. Algebra 46(2018), no. 11, 4787–4802.CrossRefGoogle Scholar
Elhamdadi, M. and Nelson, S., Quandles—an introduction to the algebra of knots. In: Student Mathematical Library, Vol. 74, American Mathematical Society, Providence, RI, 2015.Google Scholar
Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2) 126(1987), no. 2, 335–388.CrossRefGoogle Scholar
Joyce, D., A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1982), no. 1, 37–65.CrossRefGoogle Scholar
Juyumaya, J. and Lambropoulou, S., An invariant for singular knots. J. Knot Theory Ramificat. 18(2009), no. 06, 825–840.CrossRefGoogle Scholar
Matveev, S. V., Distributive groupoids in knot theory. Mat. Sb. (N.S.) 119(1982), no. 1, 78–88.Google Scholar
Nelson, S., A polynomial invariant of finite quandles. J. Algebra Appl. 7(2008), no. 2, 263–273.CrossRefGoogle Scholar
Nelson, S., Generalized quandle polynomials. Canad. Math. Bull. 54(2011), no. 1, 147–158.CrossRefGoogle Scholar
Oyamaguchi, N., Enumeration of spatial 2-bouquet graphs up to flat vertex isotopy. Part B. Topol. Appl. 196(2015), 805–814.CrossRefGoogle Scholar
Vassiliev, V. A., Cohomology of knot spaces. In: Theory of singularities and its applications, Adv. Soviet Math., 1, American Mathematical Society, Providence, RI, 1990, pp. 23–69.CrossRefGoogle Scholar